Holomorphic feedforward networks

TL;DR

This paper introduces a novel class of feedforward neural networks based on holomorphic line bundles on complex manifolds, exploring their approximation capabilities and drawing parallels with Ricci flat Kähler metric computations.

Contribution

It presents a new theoretical framework linking holomorphic geometric structures with neural network models, expanding understanding of their approximation power.

Findings

Establishes formal similarities between neural network approximation and Ricci flat Kähler metric problems.

Proposes a new class of neural networks inspired by complex geometry.

Discusses potential implications for machine learning and geometric analysis.

Abstract

A very popular model in machine learning is the feedforward neural network (FFN). The FFN can approximate general functions and mitigate the curse of dimensionality. Here we introduce FFNs which represent sections of holomorphic line bundles on complex manifolds, and ask some questions about their approximating power. We also explain formal similarities between the standard approach to supervised learning and the problem of finding numerical Ricci flat K\"ahler metrics, which allow carrying some ideas between the two problems.